A fast algorithm for quadratic resource allocation problems with nested constraints

Uiterkamp, Martijn H. H. Schoot; Hurink, Johann L.; Gerards, Marco E. T.

doi:10.1016/j.cor.2021.105451

Cited by 7 publications

(12 citation statements)

References 32 publications

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“…Such monotonicity arguments have been primarily studied for resource allocation problems where the objective function is separable, i.e., can be written as the sum of single-variable functions. However, in previous work [38] we prove the validity of similar monotonicity arguments to solve a nonseparable resource allocation problem with so-called generalized bound constraints. Moreover, recent results on the use of interior-point methods for nested resource allocation problems [40,45] suggest that incorporating specific nonseparable terms in the objective function does not increase the complexity of the used solution method.…”

Section: Discussionmentioning

confidence: 85%

“…It would be worthwhile to conduct a more thorough comparison and to develop an automated procedure to decide which algorithm is most likely to be faster. Moreover, the nonseparable version of the studied problem mentioned in the previous paragraph is related to energy management of batteries in three-phase distribution networks, where load profile flattening on all three phases together is required to avoid blackouts in these networks [44,22,38]. Thus, research in this direction is also relevant in the context of DEM.…”

Section: Discussionmentioning

confidence: 99%

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A fast algorithm for quadratic resource allocation problems with nested constraints

Uiterkamp,

Hurink,

Gerards

2020

Preprint

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We study the quadratic resource allocation problem and its variant with lower and upper constraints on nested sums of variables. This problem occurs in many applications, in particular battery scheduling within decentralized energy management (DEM) for smart grids. We present an algorithm for this problem that runs in O(n log n) time and, in contrast to existing algorithms for this problem, achieves this time complexity using relatively simple and easy-to-implement subroutines and data structures. This makes our algorithm very attractive for real-life adaptation and implementation. Numerical comparisons of our algorithm with a subroutine for battery scheduling within an existing tool for DEM research indicates that our algorithm significantly reduces the overall execution time of the DEM system, especially when the battery is expected to be completely full or empty multiple times in the optimal schedule. Moreover, computational experiments with synthetic data show that our algorithm outperforms the currently most efficient algorithm by more than one order of magnitude. In particular, our algorithm is able to solves all considered instances with up to one million variables in less than 17 seconds on a personal computer.

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Section: Discussionmentioning

confidence: 85%

Section: Discussionmentioning

confidence: 99%

A fast algorithm for quadratic resource allocation problems with nested constraints

Uiterkamp,

Hurink,

Gerards

2020

Preprint

Self Cite

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“…We conclude this section with a note on whether the result of Theorem 1 could be extended also to instances that do not fall within the class I sub . We observe that Lemma 2 can be extended to broader classes of problems with, e.g., multi-dimensional stage vectors, nonlinear and non-submodular inequality constraints, and nonseparable objectives (see, e.g., [51]). This leads us to the question whether one can also extend Lemmas 3 and 4 to broader classes of problems.…”

Section: Robustness Of Oddo Against Prediction Errorsmentioning

confidence: 99%

ODDO: Online Duality-Driven Optimization

Uiterkamp,

Gerards,

Hurink

2020

Preprint

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Motivated by energy management for micro-grids, we study convex optimization problems with uncertainty in the objective function and sequential decision making. To solve these problems, we propose a new framework called "Online Duality-Driven Optimization" (ODDO). This framework distinguishes itself from existing paradigms for optimization under uncertainty in its efficiency, simplicity, and ability to solve problems without any quantitative assumptions on the uncertain data.The key idea in this framework is that we predict, instead of the actual uncertain data, the optimal Lagrange multipliers. Subsequently, we use these predictions to construct an online primal solution by exploiting strong duality of the problem. We show that the framework is robust against prediction errors in the optimal Lagrange multipliers both theoretically and in practice. In fact, evaluations of the framework on problems with both real and randomly generated input data show that ODDO can achieve near-optimal online solutions, even when we use only elementary statistics to predict the optimal Lagrange multipliers.

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“…Applications of this problem include portfolio optimization [45], transportation problems [11], stratified sampling [67], and electric vehicle charging [69].…”

Section: α/Gbc/γ: Optimization Over Generalized Bound Constraintsmentioning

confidence: 99%

“…This problem has many applications in, e.g., machine learning [4,3], scheduling [74,44], and game theory [36,28,26]. Moreover, important special cases of this problem are the RAP with box constraints (see [61]), the RAP with generalized bound constraints (see [69]), and the RAP with nested constraints (see [82]). Important application areas for in particular these special cases include, among many others, regularized learning [12,47], telecommunications and energy management [60,79,88], and statistics [57,15] (see also the overviews in [61] and [82]).…”

Section: Introductionmentioning

confidence: 99%

On a reduction for a class of resource allocation problems

Uiterkamp¹,

Gerards²,

Hurink³

2020

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In the resource allocation problem (RAP), the goal is to divide a given amount of resource over a set of activities while minimizing the cost of this allocation and possibly satisfying constraints on allocations to subsets of the activities. Most solution approaches for the RAP and its extensions allow each activity to have its own cost function. However, in many applications, often the structure of the objective function is the same for each activity and the difference between the cost functions lies in different parameter choices such as, e.g., the multiplicative factors. In this article, we introduce a new class of objective functions that captures the majority of the objectives occurring in studied applications. These objectives are characterized by a shared structure of the cost function depending on two input parameters. We show that, given the two input parameters, there exists a solution to the RAP that is optimal for any choice of the shared structure. As a consequence, this problem reduces to the quadratic RAP, making available the vast amount of solution approaches and algorithms for the latter problem. We show the impact of our reduction result on several applications and, in particular, we improve the best known worst-case complexity bound of two important problems in vessel routing and processor scheduling from O(n 2 ) to O(n log n).

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A fast algorithm for quadratic resource allocation problems with nested constraints

Cited by 7 publications

References 32 publications

A fast algorithm for quadratic resource allocation problems with nested constraints

A fast algorithm for quadratic resource allocation problems with nested constraints

ODDO: Online Duality-Driven Optimization

On a reduction for a class of resource allocation problems

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